The documentary gives viewers a front-row seat to Edward’s Snowden’s revelations about the NSA as they unfolded day by day

The great documentaries of the last 20 years have had several things in common. Uncanny timing. Uncommon access. Footage that couldn’t be replicated. And a solid narrative arc. Those films, which would include “Crumb,” “Senna,” “Hoop Dreams” and “Man on Wire,” were also directed by people with an instinct for a good story, and the flexibility of mind to follow where it led.

With “Citizenfour”—certain to be shorthanded as “the Edward Snowden film”—director Laura Poitras has made a nonfiction thriller about the National...